Parameterized Analogues of Probabilistic Computation

Abstract

We study structural aspects of randomized parameterized computation. We introduce a new class W[P]-PFPT as a natural parameterized analogue of PP. Our definition uses the machine based characterization of the parameterized complexity class W[P] obtained by Chen et.al [TCS 2005]. We translate most of the structural properties and characterizations of the class PP to the new class W[P]-PFPT.We study a parameterization of the polynomial identity testing problem based on the degree of the polynomial computed by the arithmetic circuit. We obtain a parameterized analogue of the well known Schwartz-Zippel lemma [Schwartz, JACM 80 and Zippel, EUROSAM 79].Additionally, we introduce a parameterized variant of permanent, and prove its #W [1] completeness.

The challenges of unbounded treewidth in parameterised subgraph counting problems

“…Subject to the assumption that W [1] is not equal to FPT under randomised parameterised reductions (as described in Section 2.2), this gives the following immediate corollary. Recently, more refined models of parameterised random complexity have been developed by Montoya and Müller [39] and Chauhan and Rao [11], in which the use of randomness is restricted. However, given that the definition of an FPTRAS does not place any restrictions on the nature of the randomised algorithm, the existence of an FPTRAS for a problem does not have immediate implications in this new framework.…”

“…It was noted in Section 3.3 that recent developments in the theory of parameterised random complexity classes [39,11] are based on a more restricted notion of parameterised randomisation than is allowed in the definition of an FPTRAS. An interesting general direction for future research in parameterised counting complexity, therefore, would be to develop an alternative notion of approximability for parameterised counting problems that places corresponding restrictions on the permitted use of randomness; this would allow more precise conclusions to be drawn about the relationship between the approximability of counting problems and the complexity of the corresponding decision problems.…”

Probabilistic Parameterized Polynomial Time

“…In [12] it is shown that rejection sampling (which is forward sampling and dismissing the outcome if it does not agree with e) provides an algorithm which places {ǫ, Pr(h), Pr(e)}-Conditional Inference in paraBPP. 3 Because Conditional Inference is itself in PP, by Theorem 1 we conclude that the parameterized problem is in pPPPT.…”

“…We also mention[3] which studies PFPT, the parameterized counterpart to PP.2 In[15,16] this class was proposed under the name FERT for fixed-error randomized tractability, intended to be reminiscent of FPT. We believe that the name PPPT is more appropriate as it calls into mind the class PP as well as ppt-reductions.…”

Enumeration in Incremental FPT-Time

“…In a recent investigation, Goldberg and Papadimitriou introduced a rich theory around this complexity class that features also several aspects of proof theory [20]. Also, the investigations of Capelli and Strozecki [5] on probabilistic classes might yield further connections to the enumeration se ing via the parametrised analogues of probabilistic computation of Chauhan and Rao [6]. Furthermore, Fichte et al [17] study the parametrised complexity of default logic and present in their work a parametrised enumeration algorithm outpu ing stable extensions.…”